Preheat oven to 350 degrees F and line 2 baking sheets with silicone mats.
Using a hand mixer, beat butter, peanut butter, and brown sugar together in a large bowl until fluffy, about 2 minutes.
Beat in vanilla and egg.
Beat in ground oats, whole oats, baking soda, and salt. Mix until well combined.
Beat in peanut butter chips.
Using a 1/4 cup measuring cup, pack dough into the 1/4 cup and turn out dough onto prepared baking sheet - no more than 6 cookies on each baking sheet. Flatten out slightly with a fork like you would a peanut butter cookie.
Bake one sheet of cookies in preheated oven until golden brown, about 15 minutes.
Remove from oven and sprinkle with Maldon sea salt, if desired, then let cookies sit for 5 minutes.
Transfer to a wire cooling rack to let finish cooling or eat immediately!
Repeat with remaining dough.
Store leftovers in an airtight container at room temperature.